ANAYTICA SOUTION OF DYNAMIC FEXURA RESPONCE OF SYMMETRIC COMPOSITE TIMOSHENKO BEAMS UNDER HARMONIC FORCES Mohammed Ali Hjaji, Hasan M. Nagiar and Ezedine G. Allaboudi Mechanical and Industrial Engineering Department University of Tripoli, ibya Emails: m.hjaji@uot.edu.ly الملخص تمت د ارسة الاستجا ة الدینام ك ة العرض ة للعت ات (الكم ارت) ذات ط قات مرك ة متماثلة ومعرضة لقوى عرض ة توافق ة مختلفة. تم اشتقاق معادلات الحركة الدینام ك ة وكذلك الشروط الحد ة ذات العلاقة وفقا لنظر ة تشوه القص من الدرجة الاولى وذلك استخدام مبدأ هاملتون. تا ثی ارت تشوه القص وقصور الدو ارن ونس ة بو سون وكذلك اتجاه الا ل اف الط قات المرك ة تم اد ارجها ص غ المعادلات الحال ة حیث تم الحصول على الحل المضبوط solution) (closed form للمعادلات الناتجة والحاكمة للاستجا ة الدینام ك ة العرض ة للحالة المستقرة لعت ات توم شنكو ) Timoshenko (beams ذات ط قات مرك ة متماثلة الا ل اف. كذلك تم إیجاد ص غ الحل المضبوط للا ه ت از ازت الدینام ك ة العرض ة المستقرة لعت ات ذات تثبیت س ط supported) (simply وأخرى ذات تثبیت كابولي.(cantilever) تم إستع ارض عض الا مثلة لتوض ح إمكان ة تطبیق ص غ الحل المضبوط للعدید من الا حمال العرض ة التوافق ة لعت ات ذات مرك ات متماثلة ا ل ط ق ا ت وزوا ا الا ل اف. أظهرت النتاي ج المتحصل علیها من هذه الد ارسة صحة ودقة الحل الحالي solution) (present وذلك من خلال مقارنتها بنتاي ج مماثلة منشورة استخدام طر قة العناصر المتناه ة element) (finite واخرى.(Eact solution) الحل الدقیق استخدام ABSTRACT The fleural dynamic response of symmetric laminated composite beams subjected to general transverse harmonic forces is investigated. The dynamic equations of motion and associated boundary conditions based on the first order shear deformation are derived through the use of Hamilton s principle. The influences of shear deformation, rotary inertia, Poisson s ratio and fibre orientation are incorporated in the present formulation. The resulting governing fleural equations for symmetric composite Timoshenko beams are solved eactly and the closed form solutions for steady state fleural response are then obtained for cantilever and simply supported boundary conditions. The applicability of the analytical closed-form solution is demonstrated via several eamples with various transverse harmonic loads and symmetric cross-ply and angle-ply laminates. Results based on the present solution are assessed and validated against other well established finite element and eact solutions available in the literature. KEYWORDS: Analytical Solution; Fleural Response; Harmonic Forces; Symmetric aminated Beams; Steady State Response. INTRODUCTION Composite laminated beams are among the most important structural components widely used in aircraft wing and fuselage structures, helicopter blades, vehicle ales, Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 17
propellant and turbine blades, ship and marine structural frames due to their ecellent features such as high strength-to-weight and stiffness-to-weight ratios. In these applications, composite laminated beams are frequently subjected to cyclic dynamic loading (e.g., harmonic ecitations). Sources of such forces include aerodynamic effects, hydro-dynamic wave motion and wind loading. Also, harmonic forces may arise from unbalance in rotating machinery and propellants and reciprocating machines. In such applications, composite beams under harmonic forces cause an undesirable vibrations and prone to fatigue failures, an important limit state when designing these composite laminated beams. Under harmonic forces, the transient component of response which is induced only at the beginning of the ecitation tends to dampen out quickly and is thus of no importance in assessing the fatigue life of a composite beam. In contrast, the steady state component of the response is sustained for a long time and is thus of particular importance in fatigue design and is the subject of the present study. Within this contet, the aim of this study is to develop an accurate and efficient solution, which captures and isolates the steady state response. The present analytical closed form solution is also able to capture the quasi-static response and predict the eigen-frequencies and eigen-modes of the composite laminated beam. While the dynamic analysis of composite laminated beams based on different beam theories was the subject of significant research studies during the past few years, but most of these studies are restricted to free vibrations of composite laminated beams. Many researchers developed and presented the analytical eact solutions and finite element techniques for free vibration response of composite laminated beams. Among them, [1] developed an eact solution based on higher-order shear deformation theory to study the free vibration behavior of cross-ply rectangular beams with arbitrary boundary conditions. Based on the transfer matri method, [] investigated the in-plane and out-ofplane free vibration problem of symmetric cross-ply laminated beams. References [3-4] presented the eact epressions for the frequency equation and mode shapes for composite Timoshenko cantilever beams. His formulation captured the effects of material coupling between bending and torsional modes, shear deformation and rotary inertia. Reference [5] studies the free vibration of composite laminated beams using a higherorder shear deformation theory. The differential quadrature method is used to obtain the numerical solution of the governing differential equations for symmetrically and antisymmetrically composite beams with rectangular cross-section and for various boundary conditions. Reference [6] presented a displacement based layerwise beam theory and applied it to laminated (0 o /90 o ) and (0 o /90 o /0 o ) beams subjected to sinusoidal load. References [7-9] developed the eact dynamic stiffness matri method free vibration analyses of arbitrary laminated composite beams based on first order shear deformation, trigonometric shear deformation and higher-order shear deformation beam theories. The effects of shear deformation, rotary inertia, Poisson s ratio, aial force and etensionalbending coupling deformations are considered in their mathematical formulations. Recently, [10] developed a two-noded C1 finite beam element with five degrees of freedom per node to study the free vibration and buckling analyses of composite crossply laminated beams by using the refined shear deformation theory. Their formulations account for the parabolical variation of the shear strains through the beam depth and all coupling coming from the material anisotropy. More recently, [11] developed a finite element model based on the first order shear deformation theory to predict the static and free vibration analyses for isotropic and orthotropic beams with different boundary conditions and length-to-thickness ratios. Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 18
It should be remarked that the previous studies are mainly focused to free vibration analysis, with no attention on studying the dynamic analysis of composite laminated beams subjected to harmonic forces. To the best of author s knowledge, no study in the literature reported an analytical closed-form solution for dynamic fleural response of composite symmetric laminated Timoshenko beams under harmonic forces. Then, the purpose of this paper is to formulate the governing field equations and boundary conditions for the problem and provide the closed form eact solutions for symmetric laminated beams of rectangular cross-sections subjected to various transverse harmonic ecitations. The present analytical closed form solution captures the effects of shear deformation, rotary inertia, Poisson s ratio and fibre orientation on quasi-static and steady state dynamic responses. The present general analytical solution is (i) appropriate and efficient in analyzing the forced bending vibration of composite laminated beams subjected to transverse harmonic ecitations, (ii) suitable to achieve simple preliminary design considerations of composite beams and (iii) used as benchmarks for checking the accuracy of the results obtained from the numerical or approimate solutions. MATHEMATICA FORMUATION Kinematics Relations A prismatic multilayered beam with length, thickness h and width b, as shown in Figure (1), is considered. In the right-handed Cartesian coordinate system ( XYZ,, ) defined on the mid-plane of the beam, the X ais is coincident with the beam ais,y and Z are coincident with the principal aes of the cross-section. Since the cross-section of the composite beam has two aes of symmetry (i.e., Y and Z aes), therefore, no coupling eists between bending and torsion responses, i.e., the present study is restricted to fleural behavior in the X Z plane. Thus, the displacements for a general point p(,) z of height z from the centroidal ais of beam based on the first order shear deformation beam theory are assumed to take the form: u ( zt,,) = u( t,) + zφ ( t,) (1) p vp ( zt,,) = 0 () w ( zt,,) = w( t,) (3) p Z Z h p(,z) w z u p ϕ w p X Y (u p,w p ) p h β u (u,w) X b Figure 1: Coordinate system and displacements Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 19
in which u( t,) and w( t,) are the aial and transverse displacements of a point p(,) z on the mid-plane in the X and Z directions, vp (,z, t ) is the lateral displacement, and φ ( t,) is the rotation of the normal to the mid-plane abouty ais, where and t are spanwise coordinate and time, respectively. Strain-Displacement Relations The strain relations of the beam associated with the small-displacement theory of elasticity are given as: u ε ε κ p = + z, and γ z wp + φ where ε = u = u is the mid-plane aial strain, κ = φ = φ is the bending curvature, and the primes denote the differentiation with respect to. Constitutive Equations of Symmetric aminated Beams The constitutive equations for a symmetric laminated beam (in which the etensional-bending coupling coefficients B ij = 0 for ij=, 1,,6 ) based on the first order shear deformation theory can be obtained by using the classical lamination theory to give: N A A A ε 11 1 16 N y A 1 A A6 0 ε yy Ny A16 A6 A66 γ y = M D11 D1 D 16 κ M y 0 D1 D D6 κ y M D y 16 D6 D 66 κ y where N, Nyare the normal forces, and Ny is the in-plane force, while M, M y and M y are the bending and twisting moments, ε (4) (5), ε yy and γ y are normal and shear strains, κ, κ y andκ y are the bending and twisting curvatures, Aij and D ij denote the etensional and bending stiffnesses, respectively, and are epressed as functions of laminate ply orientation and material properties: h ij ij h ij A, D = Q 1, z dz, (for i, j = 1,,6) (6) where Qij are the transformed reduced stiffnesses and are given by the epressions [1]: ( ) 4 4 11 = 11cos β+ 1 + 66 sin βcos β+ sin β 1 = 11 + 4 66 sin cos + 1 sin 4 + cos 4 Q Q Q Q Q ( ) β β ( β β) Q Q Q Q Q 4 4 sin ( ) sin cos cos ( ) ( ) Q = Q β+ Q + Q β β+ Q β 11 1 66 3 3 16 = 11 1 66 + 1 + 66 Q Q Q Q sinβcos β Q Q Q sin βcosβ Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 0
( ) ( ) 6 = 11 1 66 sin 3 cos + 1 + 66 sin cos 3 Q Q Q Q β β Q Q Q β β ( ) sin βcos β ( sin 4 β cos 4 β) Q = Q + Q Q Q + Q + 66 11 1 66 66 in which β is the angle between the fiber direction and longitudinal ais of the beam Figure (1), Q 11, Q 1, Q and Q66 are the stiffness constants and are given in terms of engineering elastic constants by: = ( 1 ), Q = υ E ( 1 υ υ ) = υ E ( 1 υ υ ) Q E υ υ 11 11 1 1 ( 1 ) Q = E υ1υ1, and Q66 G1 1 1 11 1 1 1 1 1 =. The present formulation captures the effect of transverse shear deformation due to bending [10], then: w Q A A A w z = 55γz = 55 + φ = 55( + φ) in which Qz is the transverse shear force per unit length, 55 13cos β 3sin h 55 h 55 (7) A = k Q dz, where Q = G + G β, k is the correlation shear factor and is taken as 5/6 to account for the parabolic variation of the transverse shear stresses, the constants E11, E are Young moduli, G1, G13, G3 are shear moduli, and υ1, υ1 are Poison ratios measured in the principal aes of the layer. The laminated composite beam is subjected to fleural forces. Then, the lateral inplane forces and moments in Y direction are negligible and set to zero, i.e., Ny = Ny = My = My = 0. In order to account for Poisson s ratios, the mid-plane strains ε yy, γ y and curvatures κ y, κ y are assumed to be nonzero. For symmetric laminated beams, the etensional response is uncoupled from the fleural response of the beam, i.e., the bending stiffness coefficients B ij are ignored ( Bij = 0 for i= j= 1,,6 ), Thus, equation (5) is written as: N A 0 ε A 0 u 11 11 = = M 0 D11 κ 0 D11 φ where D A ( A A ) + A ( A A ) = +, and ( A A A ) 1 66 6 16 6 11 A11 6 66 A D ( D D ) + D ( D D ) = +. ( D D D ) 1 66 6 16 6 11 D11 6 66 If Poisson s ratio effect is ignored, the coefficients A 11, D 11 in equation (8) are then replaced by the laminate stiffness coefficients A 11, D 11, respectively. (8) Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 1
ENERGY EXPRESSIONS The total kinetic energy T for symmetric laminated composite beam (i.e., both geometric and material symmetry with respect to the mid-surface) is given by: 1 h 1 T = ρ u 0 p w p bdzd I 0 1u Iφ I1w bd + = + + h (9) where the dot denotes the derivative with respect to time, the densities I 1 and I of the composite beam are introduced by: h m 3 3 1 = ρ ρ = h k k k 1 k k 1 k= 1 ( ) ( ) I, I 1, z dz z z, z z 3 where ρk is the mas density of the k th layer. The total strain energyu for symmetric laminated composite beam is given by: 1 U = N ε + M κ + Q γ bd o z z 0 Substitution from equations (6) and (8) into the above equation, yields: 11 11φ 55( φ φ ) (10) 0 1 U = A u + D + A w + w + bd The work done V by the applied harmonic aial and fleural forces can be written as: 0 0 (11) [ z( ) ( ) ( ) ( ) ( ) φ( )] [ ( e ) ( e )] + [ P (,t) w(,t)] + [ M (,t) φ (,t)] V = q,t w,t + q,t u,t + m,t,t d + P,t u,t z e e 0 e e 0 Epressions for Force Functions The laminated composite beam illustrated in Figure () is assumed to be subjected to distributed harmonic forces and moments within the beam: q z (,t) P z (,t) M (,t) q (,t) m (,t) X P (,t) Figure : Composite cantilever under transverse harmonic forces and moments [ ] i t q (,t), q (,t), m (,t) = q ( ), q ( ), m ( ) e Ω (1) z z and to concentrated harmonic forces and moments at beam both ends: ( ), ( ), ( ) ( ), ( ), ( ) i t P,t P,t M,t = P P M e Ω (13) z z 0 Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017
where Ω is the circular eciting frequency of the applied forces, i = 1 is the imaginary constant, qz (,t ), q(,t ) are the distributed aial and transverse harmonic forces, m (,t ) is the distributed harmonic bending moment, P (,t ), Pz (,t ) are the concentrated aial and transverse harmonic forces, M (,t) is the concentrated harmonic bending moment, all forces and moments are applied at beam ends (i.e., = 0, ). Steady State Displacement Functions Under the given applied harmonic forces and moments, the displacement functions corresponding to the steady state component of the response are assumed to take the form: [ ] iωt u(,t), w(,t), φ (,t) = U ( ), W ( ), Φ ( ) e (14) in which U( ), W( ) and Φ ( ) are the amplitudes for aial translation, bending displacement, related bending rotation, respectively. Since the present formulation is intended to capture only the steady state response of the system, the displacement fields postulated in equation (14) neglect the transient component of the response. HAMITON S VARIATIONA PRINCIPE The dynamic differential equations of motion for laminated composite beam subjected to harmonic forces can be derived using Hamilton s principle, which can be written as: t δ ( T U + V ) dt = 0, where δ u = δ w= δφ = 0 at t = t1and t t (15) 1 where t 1 and t are two arbitrary time variables and δ denotes the first variation. From equations (1-14) and by substituting into the energy epressions (9-11), then, the resulting equations are substituted into Hamilton s principle (15), performing integration by parts, the following governing equations of motion are obtained: 11 1 AU ( ) IΩ U ( ) = q( ) b (16) 1 55 55 I Ω W( ) A W ( ) A Φ ( ) = q ( ) b (17) ( ) 55 11 55 z A W ( ) D Φ ( ) + A I Ω Φ ( ) = m ( ) b (18) The related boundary conditions arising from the variational principle are: 11 δ 0 0 55[ ( ) ( )] ( ) ( ) δ 0 z 0 11Φ ( ) δφ ( ) = ( ) 0 0 ba U ( ) U( ) = P ( ) (19) ba W +Φ W = P (0) bd M (1) Equation (16) governs the longitudinal deformation of the symmetric laminated composite beam which is uncoupled with the remaining equations and can be solved independently. Equations (17-18) with related boundary conditions in (0-1) provide the fleural vibration and related rotation for symmetric laminated beams. The present study Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 3
is focused on the analytical solution for the steady state dynamic response governed by the fleural equations. ANAYTICA COSED-FORM SOUTION FOR FEXURA RESPONSE Homogeneous Solution The homogeneous solution of the fleural equations (17-18) is obtained by setting the right-hand side of the equations to zero, i.e. qz( ) = m( ) = 0. The homogeneous solution of the displacements is then assumed to take the eponential form: { χ ( )} h ( ) ( ) W c = = e, for i= 1,,3,4 1 Φ h 1, i m i h c 1, i 1 where χ ( ) W ( ) ( ) h 1 h h 1 and 1 1, i, i 1 () = Φ is the vector of fleural displacement and rotation, C = c c is the vector of unknown integration constants. From equation (), by substituting into equations (17-18), yields: ( ) A mi ( A I D mi ) I Ω + A m A m c = 1 55 i 55 i m e i 0 1, i m 55 55 11 0 i c Ω e, i 1 For a non-trivial solution, the determinant of the bracketed matri in equation (3) is set to vanish, leading to the quartic equation of the form: 4 55 11 i 55 1 11 i 1 55 A D m +Ω ( A I + I D ) m +Ω I ( I Ω A ) = 0 which is depend upon section properties, material constants and eciting frequency. The above equation has the following four distinct roots; { 0} 1 (3) 1 m A I D I A D I A I D I 1, =± ( 55 11 1) 4 55 11 1 ( 55 11 1) A55D Ω + + Ω +Ω 11 1 m i A I D I A D I A I D I 3,4 =± ( 55 11 1) 4 55 11 1 ( 55 11 1) A55D Ω + + Ω +Ω 11, and For each root m i, there corresponds a set of unknown constants C = c1, i c, i. i,1 i,1 By back-substitution into equation (3), one can relate constants c 1,i to c,i c = Gc, where Gi = A55m i I1Ω + A55m i 1, i i, i, for i = 1,,3,4. The homogeneous solutions for the fleural displacement and Wh ( ) rotation Φ ( ) are obtained as: h { ( )} ( ) { } h = G E C χ 1 4 44 41 through related bending (4) Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 4
1 3 4 where G G G G G = 4, 1 1 1 1 E ( ) is a diagonal matri consisting of 44 4 m the eponential functions e i (for i= 1,,3,4 ), the vector of unknown integration constants C c,1 c 1, c,3 c 4,4 1 4 of the problem. = is to be determined from the boundary conditions Particular Solution for Uniform oad Distribution For a composite laminated beam subjected to distributed transverse forces and iωt iωt moments( qz( ),m( ) ) e = ( q z,m) e, the corresponding particular solution of the coupled bending equations (17-18) is given as: χ q z P = W 1 p Φ = p 1 bi1ω b( A55 I Ω ) 1 m The complete closed-form steady state solution for the system of fleural coupled equations is then obtained by adding the homogeneous part in equation (4) to the particular part in equation (5) to yield: { p } { χ( ) } = G E ( ) { C} + χ ( ) 1 4 44 41 1 (5) (6) Eact Solution for Cantilever Composite Beam under Transverse Harmonic Forces A cantilever composite beam subjected to (i) concentrated end harmonic forces; transverse force Pz ( e ) iω t, bending moment M( e ) iω t, and (ii) distributed harmonic i t forces; transverse force qe z Ω i t and bending moment me Ω is considered as illustrated in Figure (3). q z (,t) P z (,t) M (,t) X Figure 3: Composite cantilever beam under transverse harmonic forces and moments Imposing the following cantilever boundary conditions at beam both ends, i.e., =0, : δ W (0) = 0 (7) δφ (0) = 0 (8) ( ) ba 55 W( ) +Φ ( ) = Pz( ) (9) bd11 Φ ( ) = M( ) (30) Substitution the displacement functions in equation (6) into the boundary conditions (7-30), the total closed form solution for a cantilever symmetric laminated beam becomes: Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 5
1 { χ ( )} ( ) [ ] { } { } c = G E Ψ c Q c + χ p (31) 1 4 44 44 41 1 where Qc = Wp Φ Pz( ) A55bΦ M( ), and 14 [ Ψ ] = 1 i ( + 1) p T m m T i c G 44 i ba55e migi bd11mie 44 Eact Solution for Simply Supported Composite Beam under Transverse Harmonic Forces A simply supported composite beam subjected to (i) distributed harmonic forces: i t transverse force qe z Ω i t, bending moments me Ω, and (ii) end harmonic bending moments M( e) e iωt at beam both ends ( e = 0, ) is considered as shown in Figure (4). p. 14 M (0,t) Z q z (,t) M (,t) m (,t) X Figure 4: Simply supported composite beam under harmonic forces and moments For simply supported beam, the boundary conditions are: δ W (0) = 0 (3) bd11 Φ (0) = M (0) (33) δ W( ) = 0 (34) bd11 Φ ( ) = M ( ) (35) From equation (6), by substituting into the boundary conditions (3-35), the general analytical solution for simply-supported composite beam having symmetric laminates is obtained as: 1 { χ ( )} ( ) [ ] { } { } s = G E Ψ S Q s + χ p where (36) 1 4 44 44 41 1 Q = W M (0) W M ( ), and s 14 p p 14 T [ ] m i m i S Gi bmid11 Gie bmie D11 Ψ =. 44 44 T NUMERICA EXAMPES AND DISCUSSION While the analytical closed-form solution developed in the present study provides the steady state dynamic fleural response of symmetric laminated composite beams under various transverse harmonic forces, it can also approach the quasi-static fleural response under the given harmonic forces when adopting a very low eciting frequency Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 6
Ω 0.01ω1 compared to the first fleural natural frequency 1 ω of the system. To validate the accuracy and applicability of the present analytical solution, three eamples are presented for cantilever and simply-supported composite beams with symmetrical crossply and angle-ply laminates. The results based on the present formulation are compared with available eact solutions in the literature and established Abaqus finite shell element. In Abaqus shell model, the shell S4R element has si degrees of freedom at each node (i.e., three translations and three rotations). The S4R shell element captures the transverse shear deformation and distortional effects. Eample 1 - Symmetric aminated Beam under Distributed Transverse Harmonic Force This eample is presented in order to demonstrate the accuracy and validity of the present analytical solution to capture the quasi-static and to predict the natural frequencies and mode shapes of the composite symmetric laminated beams under transverse harmonic forces. The quasi-static response of the composite beam under harmonic forces is captured by using very low eciting frequency ( Ω 0.01ω1 ) compared to the first natural frequency ω 1 of the system. For comparison, three-layered symmetric cross-ply o o o (0,90,0) laminated composite beams for both clamped-free and simply supported boundary conditions under a uniformly distributed transverse harmonic force iωt q (, t) = 00 e N/ mare considered. z Quasi-static Response Validation For static response, the three laminates have the same thickness and made of the orthotropic composite material properties: E11= 5E, G1 = G13 = 0.5E, G3 = 0.E, 3 υ 1 = 0.5 and ρ = 1350 kg / m. For the comparison purpose, the transverse displacement function W for symmetric cross-ply laminated beam based on the present analytical solution are given in the following non-dimensional form [10] as: 3 4 W = 100bh EW qz and are compared with eact static solutions given by [10, 13] and [14]. Table (1) provides the non-dimensional mid-span transverse displacements W( = /) for clamped and simply supported symmetric composite beams under distributed transverse forces for different span-to height ratio of (/h)= 5,10,0 and 50. It is noted that the results obtained by the present formulation are in ecellent agreement with results based on other eact solutions. Table 1: Static results for symmetric cross-ply beam under distributed harmonic force Beam type Cantilever Simplysupported W( = /) (in mm) Reference (/h)=5 (/h)=10 (/h)=0 (/h)=50 [10] 6.703 3.38.485.48 [13] 6.693 3.31 -.4 [14] 6.698 3.33 -.43 Present 6.700 3.34.479.43 [10].148 1.03 0.74 0.663 [13].145 1.00-0.660 [14].146 1.01-0.661 Present.147 1.0 0.740 0.661 Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 7
Fleural Natural Frequencies Validation o o o The fundamental natural frequencies of symmetric cross-ply (0,90,0) laminated beams having clamped-free and simply supported boundary conditions are investigated. The steady state analyses of the composite beams under the given harmonic transverse iωt force qz (, t) = 00 e N/ m are solved in order to etract the fundamental transverse natural frequency of the given beam. For comparison, the following orthotropic composite material properties used are given [10] as: E11= 40E, G1 = G13 = 0.60E, 3 G3 = 0.50E, υ 1 = 0.5, ρ = 1389 kg / m. The non-dimensional natural frequencies ω etracted from the transverse steady state responses presented in Table () are conducted based on the present closed form solution, and other results available in the literature survey. o o o Table : Non-dimensional fundamental natural frequencies for symmetric (0,90,0) beam Beam type Cantilever Simply-supported Reference /h 5 10 0 [1] 4.34 5.495 - [10] 4.48 5.493 6.063 [15] 4.30 5.491 - Present 4.17 5.479 6.07 [1] 9.08 13.61 - [10] 9.94 13.6 16.33 [15] 9.07 13.61 - Present 9.0 13.64 16.36 Table () shows the first non-dimensional natural frequencies ω for span-to-height ratio (/h)= 5, 10 and 0, where the non-dimensional form is defined by: ω= ( ω h) ρ E. It is noted that the results predicted by the present analytical solution are in ecellent agreement with the corresponding results obtained from the eact solution of [1] and finite element solutions of [10] and [15]. Eample - Simply-Supported Beam under Distributed Harmonic Transverse Force A 6000mm span simply-supported composite beam having four-layered symmetric o o o o angle-ply (0, + 45, + 45,0 ) laminates subjected to uniformly distributed transverse iωt harmonic force qz (, t) = 4.0 e kn / m is considered as shown in Figure (5). The beam cross-section has width b = 00mm and thickness h = 40mm. The composite material properties used (taken from Jun et al. 008), are given as: E 11 = 41.5GPa, E = 18.98GPa, G 1 = G 13 = 5.180GPa, G 3 = 3.45GPa, 3 υ 1 = 0.4 and ρ = 015 kg / m. It is required to (a) conduct a quasi-static analysis by using very low eciting frequency Ω 0.01ω 1 related to the first natural frequency, (b) determine the steady state response at eciting frequency Ω= 1.68ω1, where the first natural frequency for the beam is Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 8
ω 1 = 30.31Hz, and (c) study the effects of fiber orientation on both quasi-static and steady state dynamic responses at Ω= 1.3ω1. Z q z (,t)=4e iωt Z kn/m =6.0m X h b Y Figure (5): Simply supported laminated beam under distributed harmonic force The present static and dynamic results are compared with Abaqus shell model solution. In Abaqus model, a total of 960 S4R shell elements are used (i.e., 6 elements along the width and 160 elements along the longitudinal ais of the beam. Static Fleural Response Based on the present formulation (i.e., equation 36), the static response of the o o o o simply-supported symmetric angle-ply (0, + 45, + 45,0 ) laminated beam under given distributed harmonic transverse force with very low eciting frequency Ω 0.01ω 1 = 0.3031Hz is approached. The quasi-static results for the maimum transverse displacement Wma at the mid-span of the beam and related bending rotation angle Φma at = 0, are presented in Table (3). It is noted that, the static response results based on the present analytical formulation are in ecellent agreement with those of the Abaqus finite element model. Table 3: Static and dynamic results for simply-supported symmetric laminated o o o o (0, + 45, + 45,0 ) beam under transverse harmonic force Response Static Ω= 0.01ω 1 Steady state Ω= 1.68ω 1 Variable Present Abaqus Solution %Difference Solution [1] =[1-]/1 [] W ma (mm) -8.580-8.496 0.98% Φ (10-3 rad) ma 4.63 4.198 1.5% W ma (mm) 4.813 4.880-1.39% Φ (10-3 rad) ma -.48 -.71-1.0% Dynamic Fleural Response iωt Under uniformly distributed transverse harmonic force qz (, t) = 4.0 e kn / m with Ω= 1.68ω1 = 50.9 Hz, the maimum amplitudes of the transverse displacement Wma and related bending rotation Φ ma for the steady state response of simplysupported symmetric (0, + 45, + 45,0 ) angle-ply laminated beam are provided in Table o o o o (3). Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 9
Figure (6a-b) and Figure (6c-d) show transverse displacement W( ) and bending rotation Φ ( ) along the beam span for quasi-static and steady state responses, respectively. The results given in tabular (Table 1) and graphical forms show an ecellent agreement between the predictions of the bending response results based on the present solution and the results of Abaqus shell model. Figure 6: Quasi-static and steady state responses for simply-supported symmetric (00 oo, +4444 oo, +4444 oo, 00 oo ) laminated beam Effect of Fiber Orientation on Composite Beam Deformations The effects of fiber orientation on the static and steady state dynamic bending responses of simply-supported symmetric composite beams under distributed transverse harmonic force are considered. The analyses are performed for quasi-static at Ω 0.01ω1 = 0.3031Hz and steady state dynamic response at eciting frequency Ω = 1.3ω = 40.0 Hz for composite beam having four-layered symmetrically o 1 o (0 / + β/ + β/0 ) angle-ply laminates in which the outer layers are kept at 0 o while the fiber angle β of the inner layers are increased ranging from β = 0 o to90 o in increments of 15 o. Table (4) provides the maimum transverse displacement at the mid-span of simply o supported symmetric (0 / + β ) s angle-ply laminated beam for static and steady state responses. As a general observation, the present analytical solution yields results in very good agreements with the corresponding results given by Abaqus shell model solution. The % difference between the two solutions (given in the last column of Table 4) are arising from the discretization errors commonly used in the Abaqus finite elements. It is noted that, as the fiber angle β increases, the maimum transverse deflections are increased for static response and decreased for the case of steady state dynamic response. It can be remarked that the laminate lay-up and stacking sequence plays the most significant role in determining the dynamic response of the composite symmetric laminated beams. Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 30
TABE 4: Static and dynamic responses for symmetric simply supported beam of different o stacking sequence laminates (0 / + β ) s Maimum fleural deflection W ma (mm) ayup Static Response Steady State Response % Difference ( β ) Ω 0.01ω1 Ω = 1.3ω1 = [1-]/1 Abaqus Present Abaqus Present Static Steady State [1] [] [1] [] 0 o -7.719-7.767 13.81 13.86-0.6% -0.36% 15 o -7.975-7.86 13.09 13.41 1.4% -.44% 30 o -8.353-8.17 1.18 1.46 1.63% -.30% 45 o -8.599-8.396 11.70 1.0.36% -.74% 60 o -8.737-8.633 11.45 11.71 1.19% -.7% 75 o -8.803-8.77 11.35 11.56 0.35% -1.85% 90 o -8.817-8.815 11.3 11.41 0.0% -0.80% Eample 3 - Cantilever Symmetric aminated Beam under End Transverse Harmonic Force A composite one end cantilever symmetric laminated beam with the same orthotropic composite material properties as that given in Eample subjected to concentrated transverse harmonic force (, ) 8.0e i Ω Pz t = t kn applied at the free end of the cantilever beam is considered as shown in Figure (7). The length of the cantilever beam is 4000mm, while the width and thickness of the rectangular cross-section are assumed equal b= h= 0.794m. The dynamic analyses are performed for cantilever beam having o o o o four-layered symmetric cross-ply (90,0,0,90) laminates. It is required to etract the natural frequencies and steady state bending modes. Abaqus model solution based on S4R shell element (i.e., 6 elements along the width and 10 elements along the longitudinal ais of the beam) are presented for comparison. P z (,t)=8.0e iωt kn =4.0m Figure 7: Cantilever symmetric laminated beam under end transverse harmonic force Etracting Bending Natural Frequencies Under the given harmonic force (, ) 8.0e i Ω Pz t = t kn, the first four natural frequencies related to the bending response are etracted from the steady state dynamic analysis when the eciting frequency Ω is varied from nearly zero to 400Hz. Figure (8) ehibits the four peaks which correspond to the first four bending natural frequencies of the cantilever beam in Table 4 for the present analytical and Abaqus shell model solutions. For the lower frequencies, both solutions closely predict the location of peaks associated with the first three natural frequencies. For higher frequencies, some discrepancy in the location of the peaks are observed between the two solutions. The present analytical X Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 31
solution yields slightly higher values than those based on the Abaqus shell model. The frequencies predicted by the present solution differed from 0.% to 1.94% from those based on Abaqus model. Figure 8: Natural frequencies for a cantilever symmetric (9999 oo, 00 oo, 00 oo, 9999 oo ) laminated beam Table 4: First four natural frequencies for cantilever symmetric (9999 oo, 00 oo, 00 oo, 9999 oo ) laminated beam under end transverse harmonic force Fleural Mode Bending natural frequencies (Hz) Abaqus Solution [1] Present Solution [] %Difference=[1-]/1 1 13.41 13.44-0.% 77.46 78.30-1.08% 3 193.9 196.9-1.55% 4 335.8 34.3-1.94% Steady State Bending Modes The dynamic responses of a cantilever symmetric cross-ply laminated o o o (90,0,0,90) o composite beam are investigated for different values of frequency ratios Ω i ω 1, i.e., applied load frequency Ωi to the first natural frequency ω 1. Figure (9) presents the first five steady state bending modes of the cantilever subjected to the given concentrated harmonic transverse force for five values of frequency ratios Ω 1= 0.50ω1, Ω = 4.50ω1, Ω 3= 7.50ω1, Ω 4 = 0.50ω1and Ω 5 = 7.50ω1, respectively. Normalized fleural displacement W/W ma 1.0 Mode 3 0.80 Mode 4 Mode 0.40 0.00-0.40-0.80 Mode 5 Mode1-1.0 (Ω 1 /ω 1 )=0.50 (Ω /ω 1 )=4.50 (Ω 3 /ω 1 )=7.50 (Ω 4 /ω 1 )=0.5 (Ω 5 /ω 1 )=7.5-1.60 0.0 0.5 1.0 1.5.0.5 3.0 3.5 4.0 (b) Beam coordinate (m) Figure 9: Natural frequencies and bending modes for a cantilever symmetric (90 oo, 0 oo, 0 oo, 90 oo ) laminated composite beam Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 3
CONCUSION From the results obtained in this study, it can be concluded that: Based on the first order shear deformation theory, the dynamic equations of motion for fleural vibration and related boundary conditions for composite symmetric laminated beams subjected to various transverse harmonic forces are derived via Hamilton s variational principle. Eact epressions for closed-form solutions of bending equations are obtained for cantilever and simply supported symmetric composite beams. The present analytical solutions are efficient in capturing the quasi-static and steady state dynamic responses of composite symmetric cross-ply and angle-ply laminated beams under different transverse harmonic forces. It is also capable of etracting the natural frequencies and steady state bending modes. Comparisons with established Abaqus finite element shell solution and eact solutions available in the literature demonstrate the validity and accuracy of the present analytical solution. REFERENCES [1] Khdeir, A. A. and Reddy, J. N., Free vibration of cross-ply laminated beams with arbitrary boundary conditions, International Journal of Engineering Science, 3 (1994), 1971-1980. [] Yildirim, V., Sancaktar, E., Kiral, E., Free vibration analysis of symmetric crossply laminated composite beams with the help of the transfer matri approach, Communications in Numerical Methods in Engineering, 15 (9), (1999), 651-660. [3] Banerjee, J. R., Frequency equation and mode shape formulae for composite Timoshenko beams, Composite Structures, 51 (001), 381-388. [4] Banerjee, J. R., Eplicit analytical epressions for frequency equation and mode shape of composite Timoshenko beams, International Journal of Solids and Structures, 38 (001), 415-46. [5] ijuan, F., et al., Free vibration analysis of laminated composite beams using differential quadrature method, Tsinghua Science and Technology, 7 (6), (00), 567-573. [6] Tahani, M., Analysis of laminated composite beams using layerwise displacement theories, Composite Structures, 79 (007), 535-347. [7] Jun,., Honging, H. and Rongying, S., Dynamic finite element method for generally laminated composite beams, 50 (008), 466-480. [8] Jun,. and Honging, H., Dynamic stiffness analysis of laminated composite beams using trigonometric shear deformation theory, Composite Structures, 89 (009), 433-44. [9] Jun,., and Honging, H., Free vibration analysis of aially loaded composite beams based on higher-order shear deformation, Meccanica, 46 (011), 199-1317. [10] Vo, T., and Inam, F., Vibration and Buckling of Cross-Ply Composite Beams using Refined Shear Deformation Theory, nd International Conference on Advanced Composite Materials and Technologies for Aerospace Applications, (01), Glyndŵr University. Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 33
[11] Elshafei, M. A., FE Modeling and Analysis of Isotropic and Orthotropic Beams Using First Order Shear Deformation Theory, Materials Sciences and Applications, 4 (013), 77-10. [1] Jones, R. M., Mechanics of Composite materials, 1975, McGraw-Hill, New York. [13] Chakraborty, A. D., et al., Finite element analysis of free vibration and wave propagation in asymmetric composite beams with structural discontinuities, Composite Structures, 55 (1), (00), 3-36. [14] Khdeir, A. and Reddy, J. N., An eact solution for the bending of thin and thick cross-ply laminated beams, Composite Structures, 37(1997), 195-03. [15] Murthy, M. V. et al., A refined higher order finite element foe asymmetric composite beams, Composite Structures, 67 (005), 7-35. NOMENCATURES A Cross-sectional area A 55 Coefficient b Width of the beam cross-section D 11 Bending stiffness constant E Modulus of elasticity h Height of the beam cross-section G Shear modulus I1, I Densities of the composite material J Torsional constant ength of the composite beam M t, Concentrated harmonic moment ( ) (, ) (, ) (, ) m t Distributed harmonic moment z P t Concentrated harmonic transverse force qz t Distributed harmonic transverse force t Time in seconds t1, t Time intervals T Kinetic energy up, vp, wp Displacements of a point p on the cross- section along ZY,, Xaes U V XYZ,, z ρ φ Ω δ ( t, ) Internal strain energy Work done by applied forces Right-handed Cartesian coordinate system Height Density of the composite beam material Rotation of the normal to the mid-plane abouty ais Eciting frequency First variation Journal of Engineering Research (University of Tripoli, ibya) Issue (3) March 017 34